The number of orbits of reflections of a finite Cartan type. Let $W$ be the Weyl group of a Cartan type. The reflections in $W$ are closed under conjugation, and this statistic counts the number of conjugacy classes of $W$ that are reflections. It is well-known that there are either one or two such conjugacy classes.

The number of parts from which one can substract 2 and still get an integer partition.

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

The length of the longest run of ones in a binary word.

The exponent of a binary word. This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called '''frequency'''.

The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.

The number of different multiplicities of parts of an integer partition.

Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.

The length of the longest constant subword.

The number of graphs with the same chromatic polynomial.